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Boundary transitions from a single round of measurements on gapless quantum states

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Pith's one-line read A single round of post-selected measurements on a gapless parent of the cluster state creates long-range order that survives tilting the measurement basis up to a critical angle, beyond which the state crosses over to power-law…

desk verdict A genuinely new measurement-induced boundary transition in a gapless parent of the cluster state, backed by BCFT and DMRG; the main soft spot is the incomplete analytic control of the Z2-breaking intermediate fixed point. read the letter →

arxiv 2412.07830 v1 pith:UWCZUDVX submitted 2024-12-10 quant-ph cond-mat.stat-mechcond-mat.str-elhep-th

Yue Liu , Sara Murciano , David F. Mross , Jason Alicea This is my paper · ORCID
classification quant-phcond-mat.stat-mechcond-mat.str-elhep-th
keywords measurement-inducedboundarytransitiongaplessparentofclusterstateconformalfieldtheoryweakmeasurementLuttingerliquidKennedy-TasakidualitytricriticalIsingmodelthree-statePotts
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that measurements alone, without Hamiltonian dynamics, can drive qualitative transitions in gapless quantum matter once the measurement basis is tilted. The authors construct a gapless parent of the one-dimensional cluster state that maps to two decoupled Luttinger liquids, and show that measuring one chain with post-selection for uniform outcomes turns the other chain long-range ordered while preserving power-law correlations. Rotating the measurement basis away from this axis preserves the long-range order up to a critical tilt angle, then gives way to power-law correlations: a measurement-induced boundary transition between distinct boundary conformal field theory fixed points. They demonstrate the same phenomenon in tricritical Ising and three-state Potts critical chains and propose a general criterion for such transitions.

What carries the argument

The central object is the gapless parent of the cluster state: the reflection-symmetric completion of the $ZXZ$ commuting-projector Hamiltonian, which under a Kennedy-Tasaki duality maps to two decoupled $XXZ$ chains, i.e., a two-channel Luttinger liquid with Luttinger parameter $K$. The mechanism is the measurement-induced boundary perturbation: a post-selected weak measurement of $X$ operators appears as $\delta S_{\mathrm{meas}}\propto \beta\int dx\,\delta(\tau)\cos\theta\cos\tilde{\theta}$, which in symmetric and antisymmetric field combinations becomes $\cos\theta_+ + \cos\theta_-$, each relevant for $K>1/2$ and driving renormalization-group flow to Dirichlet or Neumann boundary conditions in the boundary conformal field theory. The tilt angle $\omega$ weights the two cosines; at $\omega_c=\pi/4$ the coefficient of $\cos\theta_+$ vanishes, leaving a $c=1$ intermediate fixed point. For the tricritical Ising and three-state Potts models, the same logic operates through the boundary-condition spectrum and boundary RG flows between free, fixed, and mixed boundary conditions.

What would settle it

In the $\mathbb{Z}_2$-preserving protocol with $K=1.5$, the derivative $\frac{d}{d\omega}\langle Z_{L/4,2}Z_{3L/4,2}\rangle$ should develop a peak at $\omega_c=\pi/4$ that sharpens with system size $L$; a numerical scan that shows no sharpening, or long-range order persisting for all $\omega$, would rule out the transition. Equivalently, at $\omega=\pi/4$ the half-chain entanglement should fit $S=\frac{c_{\mathrm{eff}}}{3}\ln[(L/\pi)\sin(\pi l/L)]$ with $c_{\mathrm{eff}}\approx 1$, so a clear area-law fit would falsify the $c=1$ intermediate fixed point.

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Extended reading notes

Core claim

The central discovery is that the post-measurement state of a gapless quantum system can realize several stable boundary fixed points, and a single parameter, the tilt angle of the measurement basis, moves the system between them. For the gapless parent state, X-basis measurements in the uniform post-selection sector produce long-range Z order with area-law entanglement, coexisting with power-law X and string correlations; Z-basis measurements yield the dual correlations. Tilting between X and Z exposes an intermediate fixed point at a critical angle, $\omega_c=\pi/4$ for the $\mathbb{Z}_2$-preserving protocol, described by a $c=1$ boson boundary conformal field theory with logarithmic entanglement, separating the two stable regimes. The same measurement-induced boundary transition appears when outcomes are averaged non-linearly through a replica construction, and the phenomenon extends to tricritical Ising and three-state Potts critical points, where tilted measurements flow between free and fixed boundary conditions through unstable mixed boundary conditions.

Load-bearing premise

The load-bearing premise is that a post-selected weak measurement on the lattice is faithfully captured by a relevant boundary perturbation in the field theory, so that the renormalization-group flow to a boundary conformal field theory fixed point determines all universal correlations of the post-measurement state.

Editorial extensions

If this is right

  • Weak measurements with any nonzero strength generate the long-range order in the gapless parent, because the boundary perturbation is relevant for $K>1/2$, in contrast to the gapped cluster state where only strict projective $X$ measurement works.
  • A decoding protocol that averages over all measurement outcomes with a sign structure yields power-law $ZZ$ correlations with exponent $1/(2K)$, distinct from the pre-measurement exponent $1/2$ whenever $K\neq 1$.
  • The $\mathbb{Z}_2$-preserving tilted measurement has a transition at $\omega_c=\pi/4$ with effective central charge $c_{\mathrm{eff}}\approx 1$; the $\mathbb{Z}_2$-breaking tilted measurement also has a transition, at $\omega_c\approx 0.22\pi$ for $K=1.5$, with $c_{\mathrm{eff}}$ depending on $K$.
  • Non-linear, Born-squared averages over measurement outcomes show the same transition, so the phenomenon is not an artifact of rare post-selected trajectories.
  • Tricritical Ising and three-state Potts critical chains exhibit measurement-induced boundary transitions between free and fixed or mixed boundary conditions, supporting a general criterion: at least two stable boundary fixed points with an unstable fixed point between them.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The sharp contrast with the gapped cluster state suggests that gaplessness is the essential resource: a continuum of low-energy boundary operators supplies the competing relevant perturbations that a single tilt angle can tune between.
  • The $K$-dependent effective central charge at the $\mathbb{Z}_2$-breaking intermediate fixed point, which the paper leaves open, may indicate a line of boundary fixed points rather than an isolated transition.
  • The criterion that measurement operators with scaling dimension below $1/2$ can drive typical-outcome transitions points to concrete experimental targets, such as Rydberg-atom realizations of tricritical Ising physics, where post-selection-free versions of these transitions could be sought.
  • The same boundary-perturbation logic should apply to any gapless state whose low-energy theory supports two competing relevant operators with a symmetry forcing their coefficients to swap as the measurement basis rotates.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper studies the effect of a single round of weak or projective measurements on a gapless parent of the one-dimensional cluster state. The parent is a reflection-symmetric ZXZ ladder that maps via a Kennedy-Tasaki transformation to two decoupled XXZ chains, and the authors use this mapping together with boundary conformal field theory (BCFT) to describe post-selected uniform-outcome measurements as relevant boundary perturbations. They report three main results: (i) X-basis measurements generate long-range Z order coexisting with power-law correlations and area-law entanglement, with a decoding protocol that reveals modified power laws without post-selection; (ii) tilting the measurement basis in the XZ plane produces a measurement-induced boundary transition, exact at omega_c = pi/4 for a Z2-preserving protocol and numerically at omega_c ~ 0.22 pi for a Z2-breaking protocol at K = 1.5, with a continuously K-dependent effective central charge at the latter transition; and (iii) analogous transitions occur in the tricritical Ising and three-state Potts chains. The claims are supported by BCFT calculations and extensive DMRG/iDMRG simulations.

Significance. If the results hold, the paper establishes a general and conceptually clean mechanism by which a single round of measurements on a gapless state can induce boundary renormalization-group flows between multiple stable BCFT fixed points, producing transitions that are absent in the gapped cluster-state descendant. The construction of the gapless parent and the duality argument are elegant, and the Z2-preserving critical angle omega_c = pi/4 is obtained without any fitting. The correlation exponents in Eqs. (28), (29), (35), (36), (41) and in Tables III and IV are derived from BCFT and then checked numerically, rather than extracted from the target data. The extension to minimal models uses established Cardy-state data and yields concrete power-law predictions with numerical support. The main caveat is that the Z2-breaking intermediate fixed point is characterized mostly numerically and its K-dependent effective central charge is left unexplained; the authors acknowledge this limitation in the main text.

minor comments (5)
  1. [Sec. VIB, Tables III and IV] The stability of the omega = 0 and omega = pi/2 fixed points for K > 1 is established by checking a finite set of symmetry-allowed operators. Because the existence of the Z2-breaking transition relies on both endpoints being stable, please add a brief argument clarifying whether the listed operators are expected to exhaust the leading relevant boundary perturbations, or alternatively rephrase the text so that the stability statement is presented as numerical evidence rather than an exhaustive classification. The DMRG data at K = 1.5 and the duality-protected Z2-preserving transition mitigate the risk, so in my reading this is not a blocker.
  2. [Sec. VIB and Fig. 8(c)] The continuous K-dependence of the effective central charge at the Z2-breaking intermediate fixed point is explicitly left as an open question. Since this is the only quantitative property of that fixed point, please state clearly that ceff is an effective fitting parameter and has not yet been shown to be a true conformal central charge, and comment on what is expected as K approaches 1, where Z_{j,1} becomes marginal at the omega = 0 endpoint.
  3. [Sec. IVA] The beta << 1 relation tau* ~ beta^{4K/(1-4K)} is obtained by dimensional analysis, while only the beta >> 1 limit is directly matched to the perturbative calculation in Appendix B. Since tau* sets a crossover scale rather than the asymptotic power-law exponents, the central BCFT predictions are unaffected, but the scaling argument could be stated more explicitly to avoid the impression that the beta-dependence at intermediate strength is derived rather than inferred.
  4. [Sec. VIIB, Eq. (60)] The three-state Potts measurement operator is introduced without explaining why the (V + V^dagger) factors are inserted and how the symmetry of the operator changes with omega. A short derivation of the defect-line action and an explicit mapping of the omega intervals to the A, B, C and mixed boundary conditions would make this section much easier to follow.
  5. [Appendix F and miscellaneous text] There are a few typographical issues: 'represeatation' in the first sentence of Appendix F, 'obtained obtain' in the text near Fig. 8(c), and the subscript 'uni' first appears in figures before it is defined in the main text. These are minor and should be corrected in a final proofread.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central exponents and transitions are derived from external BCFT input and bosonization, then independently benchmarked with DMRG.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The post-selected weak measurement is converted into a boundary perturbation (Eqs. 16, 23, 44) by standard bosonization of the gapless-parent operators, and the universal correlation exponents (e.g., min(4,4K), 1/(2K), 2K, and the minimal-model exponents 3, 4, 4/3, 4/5) are read off from known boundary CFT spectra (Cardy states; Refs. 32, 56, 57, 63) or from the derived boundary action, not fitted to the post-measurement data. The DMRG/iDMRG simulations serve as an independent benchmark: they simulate the microscopic post-measurement wavefunction and reproduce the predicted exponents (Figs. 3, 16, 17, 23, 28, 29). The one analytically unprotected critical angle (omega_c ~ 0.22pi in the Z2-breaking protocol) and the K-dependent effective central charge are explicitly presented as numerical extractions, with the paper flagging the ceff(K) dependence as an open question, so they are not fitted values renamed as predictions. The omega_c = pi/4 transition in the Z2-preserving protocol follows from an exact duality and the vanishing of a coupling in Eq. (45), not from a fit. The main physical input, that a relevant tau=0 perturbation flows to the assumed BCFT fixed point, is a hypothesis rather than a tautology; its validity is supported by the independent DMRG agreement, and the paper candidly notes where support is incomplete (e.g., the ceff(K) puzzle and the partial operator census). Self-citations (Refs. 12, 13, 49, 55) are contextual or methodological and are not used to force the central results; the load-bearing BCFT data are external.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard bosonization, BCFT, and the Kennedy-Tasaki duality, plus the assumption that post-selected weak measurements are faithfully described by boundary perturbations. The only numerical extractions that are not derived analytically are the critical tilt angle and the effective central charge for the Z2-breaking measurement basis, but these do not feed back into the derivation of the transitions.

free parameters (2)
  • Critical tilt angle omega_c for the Z2-breaking measurement basis = approx 0.22 pi for K = 1.5
    Extracted numerically from DMRG data (peak in derivative of order parameter, Fig. 7) rather than derived analytically. The existence of the transition is the central claim, but the exact location for the Z2-breaking case is not predicted from theory.
  • Effective central charge ceff of the Z2-breaking intermediate fixed point = varies with K, e.g., approx 0.52 for K = 1.5
    Obtained by fitting the half-chain entanglement entropy scaling to S_L/2 = (ceff/6) log L + const (Sec. VIB, Fig. 8). The continuous dependence on K is observed but not derived analytically.
assumptions (4)
  • standard math The Kennedy-Tasaki duality exactly maps the gapless parent Hamiltonian to two decoupled XXZ chains.
    Used in Sec. III to establish the Luttinger liquid description. The duality is a known nonlocal unitary transformation, cited from Refs. [42-45].
  • domain assumption The bosonization dictionary and the Luttinger liquid action for the XXZ chain are valid in the low-energy limit.
    Standard bosonization results (Ref. [50]) are used throughout Sec. IV and appendices to map spin operators to vertex operators and to compute scaling dimensions.
  • domain assumption A post-selected weak measurement with uniform outcome is equivalent to a relevant boundary perturbation in the CFT, whose RG flow to a BCFT fixed point determines the universal properties.
    The central tool of the paper, introduced in Sec. IVA: the measurement-induced term is treated as a delta-function perturbation at tau = 0, and the resulting state is described by boundary conditions at a beta-dependent tau-star. This assumption is standard in the literature but is not rigorously derived.
  • domain assumption The perturbative expansion in u = exp(-2 beta) around the projective limit is valid when the strange correlator exponent eta > 1.
    Used in Appendix B to compute correlation functions from a truncated wavefunction. The authors note that the expansion is well-behaved only if eta > 1, which they state holds for the relevant operators studied in this work.

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Cite this review

Pith. "Pith review of Boundary transitions from a single round of measurements on gapless quantum states." pith.science (2026). https://pith.science/paper/UWCZUDVX

@misc{pith2026241207830,
  author       = {Pith},
  title        = {Pith review of: Boundary transitions from a single round of measurements on gapless quantum states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UWCZUDVX}},
  note         = {Machine review of arXiv:2412.07830}
}
read the original abstract

Measurements can qualitatively alter correlations and entanglement emerging in gapless quantum matter. We show how a single round of measurements on gapless quantum systems can, upon rotating the measurement basis, induce non-trivial transitions separating regimes displaying universal characteristics governed by distinct boundary conformal field theories. We develop the theory of such `measurement-induced boundary transitions' by investigating a gapless parent of the one-dimensional cluster state, obtained by appropriately symmetrizing a commuting projector Hamiltonian for the latter. Projective measurements on the cluster state are known to convert the wavefunction, after post-selection or decoding, into a long-range-ordered Greenberger-Horne-Zeilinger (GHZ) state. Similar measurements applied to the gapless parent (i) generate long-range order coexisting with power-law correlations when post-selecting for uniform outcomes, and (ii) yield power-law correlations distinct from those in the pre-measurement state upon decoding. In the post-selection scenario, rotating the measurement basis preserves long-range order up until a critical tilt angle marking a measurement-induced boundary transition to a power-law-ordered regime. Such a transition -- which does not exist in the descendant cluster state -- establishes new connections between measurement effects on many-body states and non-trivial renormalization-group flows. We extend our analysis to tricritical Ising and three-state Potts critical theories, which also display measurement-induced boundary transitions, and propose general criteria for their existence in other settings.

Figures

Figures reproduced from arXiv: 2412.07830 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Canonical cluster state model. The Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Phase diagram of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Correlation functions and (b) entanglement en [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (26 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Entanglement entropy scaling at the measurement [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Derivative of order and disorder parameter corre [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (a,b) Dependence of half-chain entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Graphical representation of the setup considered to [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Derivative of order and disorder parameter correla [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. (a) Boundary renormalization group flow involving [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Boundary RG flow between fixed and mixed bound [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Upper row: Schematic RG flow induced by measurements enacted by [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The strange correlator [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. BCFT and method of images. The measurement pins [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Top panel: the correlation function [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Top panel: the expectation value [PITH_FULL_IMAGE:figures/full_fig_p027_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Two-point correlation function of [PITH_FULL_IMAGE:figures/full_fig_p029_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. The derivative of order and disorder parameters in the (a) upper and (b) lower chain of the post-measurement gapless [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. The derivative of order and disorder parameters in the lower chain of the post-measurement gapless parent of the [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Correlation functions of [PITH_FULL_IMAGE:figures/full_fig_p031_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. The entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p032_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. Two-point correlation functions of selected operators in the gapless parent of the cluster state with Luttinger parameter [PITH_FULL_IMAGE:figures/full_fig_p033_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Entanglement entropy obtained by DMRG with different bond dimensions [PITH_FULL_IMAGE:figures/full_fig_p033_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Order and disorder parameters as a function of the tilting angle for (a) [PITH_FULL_IMAGE:figures/full_fig_p034_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. The correlation functions of the ground state of a spin-1 [PITH_FULL_IMAGE:figures/full_fig_p035_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Select correlation functions of the spin-1 [PITH_FULL_IMAGE:figures/full_fig_p036_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. Post-measurement correlation functions of [PITH_FULL_IMAGE:figures/full_fig_p037_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Correlation functions of three-state Potts model after measurement of [PITH_FULL_IMAGE:figures/full_fig_p038_29.png]

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    Replica model with different values ofK Since it is difficult to study analytically the replica model of Section VIC, we add more numerical results for the tilted measurement basis with different values ofK. In Fig. 25 we plot the order parameter ,⟨⟨Z L 4 ,2Z 3L 4 ,2⟩⟩, and di...

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